{
  "id": "1511.03746",
  "version": "v1",
  "published": "2015-11-12T01:17:33.000Z",
  "updated": "2015-11-12T01:17:33.000Z",
  "title": "Helicity is the only invariant of incompressible flows whose derivative is continuous in $C^1$-topology",
  "authors": [
    "Elena A. Kudryavtseva"
  ],
  "comment": "5 pages",
  "categories": [
    "math.DS",
    "math-ph",
    "math.DG",
    "math.MP"
  ],
  "abstract": "Let $Q$ be a smooth compact orientable 3--manifold with smooth boundary $\\partial Q$. Let $\\mathcal{B}$ be the set of exact 2--forms $B\\in\\Omega^2(Q)$ such that $j_{\\partial Q}^*B=0$, where $j_{\\partial Q}:{\\partial Q}\\to Q$ is the inclusion map. The group $\\mathcal{D}=\\mathrm{Diff}_0(Q)$ of self-diffeomorphisms of $Q$ isotopic to the identity acts on the set $\\mathcal{B}$ by $\\mathcal{D}\\times\\mathcal{B}\\to\\mathcal{B}$, $(h,B)\\mapsto h^*B$. Let $\\mathcal{B}^\\circ$ be the set of 2--forms $B\\in\\mathcal{B}$ without zeros. We prove that every $\\mathcal{D}$--invariant functional $I:\\mathcal{B}^\\circ\\to\\mathbb{R}$ having a regular and continuous derivative with respect to the $C^1$--topology can be locally (and, if $Q=M\\times S^1$ with $\\partial Q\\ne\\varnothing$, globally on the set of all 2--forms $B\\in\\mathcal{B}^\\circ$ admitting a cross-section isotopic to $M\\times\\{*\\}$) expressed in terms of the helicity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-12T01:17:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35",
      "37A20",
      "37C15",
      "53C65",
      "53C80"
    ],
    "keywords": [
      "incompressible flows",
      "derivative",
      "continuous",
      "smooth boundary",
      "cross-section isotopic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151103746K"
    }
  }
}